A small, startup company (with one engineer employed) needs to use a shaft made of two materials in its new machine. The company is advertising to seek an outside engineering consultant to quickly provide it with the answers (and show its engineer how to get the answers) for the following problem. If the engineering consultant cannot deliver the answers directly, the company will ask the consultant to at least provide it with a detailed method and guide so its engineer may follow step-by-step to get the answers. A prismatic shaft is made of two materials. The cross section of the two-material prismatic shaft has an outer equilateral triangle made of Material-1 (green area in Figure) and an inner square made of Material-2 (yellow area in Figure). Material-1 has shear modulus G and Material-2 has shear modulus G2. Both materials have the same Poisson's ratio, V = V = v. y - side 1 x = -2a side 2 side 3 x x = a A torque Mz is applied and the shaft deforms in torsion with an angle of twist per unit length a. A torque Mz is applied and the shaft deforms in torsion with an angle of twist per unit length a. The equations describing the 3 sides of the triangle are: x - 3y+2a = 0 (side 1) x + 3y+ 2a = 0 (side 2) x-a=0 (side 3) The equations that bound the 4 sides of the square are: x- 0.5a = 0 x+0.5a=0 y - 0.5 = :0 y + 0.5 = 0 For this two-material prismatic shaft, the company needs to find: (a) The shear stress distribution xz(x,y) and yz(x,y). (b) The torsional rigidity D. (c) The warping function x(x, y).